City, having seen off Borussia Dortmund, will now face Paris St Germain, while Real's reward for getting past Liverpool is a clash with Chelsea.
Here, the PA new agency takes a look at how the teams left in Europe's elite club competition shape up.
Paris St Germain
Group stage: Group H winners
Quarter-final result: 3-3 v Bayern Munich (A: 3-2, H: 0-1)
European Cup best: Runners-up (2020)
PSG advanced to the last four by getting the better of the side who beat them in last season's final, Bayern. The quarter-final tie featured two goals from Kylian Mbappe, who has scored eight times in the competition this term. While they are playing catch-up behind Lille in Ligue 1, Mauricio Pochettino's men look capable of causing major problems for any opposition.
Group stage: Group C winners
Quarter-final result: 4-2 v Borussia Dortmund (H: 2-1, A: 2-1)
European Cup best: Semi-finals (2016)
Manchester City survived an early scare in Germany to see off Borussia Dortmund and reach the semi-finals for the first time under Pep Guardiola. Jude Bellingham levelled up the tie after 15 minutes of the second leg at Signal Iduna Park but second-half goals from Riyad Mahrez, from the penalty spot, and Phil Foden saw City win 2-1 on the night and progress 4-2 on aggregate. The result ended City's run of quarter-final frustration, having gone out at the last-eight stage in each of the previous three years, and kept alive their hopes of a potential unprecedented quadruple.
Group stage: Group B winners
Quarter-final result: 3-1 v Liverpool (H: 3-1, A: 0-0)
European Cup best: Winners – 13 (1956, 1957, 1958, 1959, 1960, 1966, 1998, 2000, 2002, 2014, 2016, 2017, 2018)
Real Madrid swept aside Liverpool in the first leg and then defended their 3-1 lead at Anfield, where a goalless draw saw them advance. The hosts created enough chances to win but Zinedine Zidane's team rode the storm. Los Blancos may not be the force they once were, but Zidane's squad have hit a decent run of form, unbeaten since the start of February.
Group stage: Group E winners
Quarter-final result: 2-1 v Porto (A: 2-0, H: 0-1)
European Cup best: Winners (2012)
Goals from England pair Mason Mount and Ben Chilwell helped the Blues come through their last-eight tie as boss Thomas Tuchel, having overseen PSG's run last term, moved closer to another final. The 1-0 loss in the second leg against Porto was only a second defeat in 18 matches for Chelsea since Tuchel succeeded Frank Lampard in January – they are currently fifth in the Premier League, a point behind fourth-placed West Ham.